Covering dimension \(\dim \)  was originally introduced by Lebesgue for domains in Euclidean spaces [1]}.
In 1933 Čech extended Lebesgue's definition to completely regular spaces [2]}, and in 1950 Katětov
proposed a different definition, which coincided with Čech's one for normal spaces [3]}. Since then,
these two versions of covering dimension have become equally popular. Thus, the books [4]}, [5]}, and
[6]} use Čech's definition, while, e.g., [7]} prefers Katětov's one. There are
also books on dimension theory which consider only normal or even only metrizable spaces [8]},
[9]}.
